Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.ZArith.Zdigits

Bit vectors interpreted as integers. Contribution by Jean Duprat (ENS Lyon).
Require Import Bvector.
Require Import ZArith.
Require Export Zpower.
Require Import Lia.
The evaluation of boolean vector is done both in binary and two's complement. The computed number belongs to Z. We hence use lia to perform computations in Z. Moreover, we use functions 2^n where n is a natural number (here the vector length).
Section VALUE_OF_BOOLEAN_VECTORS.
Computations are done in the usual convention. The values correspond either to the binary coding (nat) or to the two's complement coding (int). We perform the computation via Horner scheme. The two's complement coding only makes sense on vectors whose size is greater or equal to one (a sign bit should be present).
#[deprecated(note="Use Z.b2z instead", since="8.18")]
Definition bit_value (b:bool) : Z :=
match b with
| true => 1%Z
| false => 0%Z
end.
#[deprecated(note="Consider Z.setbit instead", since="8.18")]
Lemma binary_value : forall n:nat, Bvector n -> Z.
#[deprecated(since="8.18")]
Lemma two_compl_value : forall n:nat, Bvector (S n) -> Z.
End VALUE_OF_BOOLEAN_VECTORS.
Section ENCODING_VALUE.
We compute the binary value via a Horner scheme. Computation stops at the vector length without checks. We define a function Zmod2 similar to Z.div2 returning the quotient of division z=2q+r with 0<=r<=1. The two's complement value is also computed via a Horner scheme with Zmod2, the parameter is the size minus one.
#[deprecated(note="Consider Z.odd or Z.modulo instead", since="8.18")]
Definition Zmod2 (z:Z) :=
match z with
| Z0 => 0%Z
| Zpos p => match p with
| xI q => Zpos q
| xO q => Zpos q
| xH => 0%Z
end
| Zneg p =>
match p with
| xI q => (Zneg q - 1)%Z
| xO q => Zneg q
| xH => (-1)%Z
end
end.
#[deprecated(note="Use Z.div2_odd instead", since="8.18")]
Lemma Zmod2_twice :
forall z:Z, z = (2 * Zmod2 z + bit_value (Z.odd z))%Z.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Z_to_binary : forall n:nat, Z -> Bvector n.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Z_to_two_compl : forall n:nat, Z -> Bvector (S n).
End ENCODING_VALUE.
Section Z_BRIC_A_BRAC.
Some auxiliary lemmas used in the next section. Large use of ZArith. Deserve to be properly rewritten.
#[deprecated(note="Consider Z.div2_odd instead", since="8.18")]
Lemma binary_value_Sn :
forall (n:nat) (b:bool) (bv:Bvector n),
binary_value (S n) ( b :: bv) =
(bit_value b + 2 * binary_value n bv)%Z.
#[deprecated(note="Consider Z.div2_odd instead", since="8.18")]
Lemma Z_to_binary_Sn :
forall (n:nat) (b:bool) (z:Z),
(z >= 0)%Z ->
Z_to_binary (S n) (bit_value b + 2 * z) = Bcons b n (Z_to_binary n z).
#[deprecated(note="Consider N.testbit instead", since="8.18")]
Lemma binary_value_pos :
forall (n:nat) (bv:Bvector n), (binary_value n bv >= 0)%Z.
#[deprecated(note="Consider Z.bits_opp instead", since="8.18")]
Lemma two_compl_value_Sn :
forall (n:nat) (bv:Bvector (S n)) (b:bool),
two_compl_value (S n) (Bcons b (S n) bv) =
(bit_value b + 2 * two_compl_value n bv)%Z.
#[deprecated(note="Consider Z.bits_opp instead", since="8.18")]
Lemma Z_to_two_compl_Sn :
forall (n:nat) (b:bool) (z:Z),
Z_to_two_compl (S n) (bit_value b + 2 * z) =
Bcons b (S n) (Z_to_two_compl n z).
#[deprecated(note="Consider Z.div2_odd instead", since="8.18")]
Lemma Z_to_binary_Sn_z :
forall (n:nat) (z:Z),
Z_to_binary (S n) z =
Bcons (Z.odd z) n (Z_to_binary n (Z.div2 z)).
#[deprecated(note="Consider Z.div2_odd instead", since="8.18")]
Lemma Z_div2_value :
forall z:Z,
(z >= 0)%Z -> (bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z.
#[deprecated(note="Use Z.div2_nonneg instead", since="8.18")]
Lemma Pdiv2 : forall z:Z, (z >= 0)%Z -> (Z.div2 z >= 0)%Z.
#[deprecated(note="Consider Z.div_lt_upper_bound instead", since="8.18")]
Lemma Zdiv2_two_power_nat :
forall (z:Z) (n:nat),
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z -> (Z.div2 z < two_power_nat n)%Z.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Z_to_two_compl_Sn_z :
forall (n:nat) (z:Z),
Z_to_two_compl (S n) z =
Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z)).
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Zeven_bit_value :
forall z:Z, Zeven.Zeven z -> bit_value (Z.odd z) = 0%Z.
#[deprecated(note="Use Zodd_bool_iff instead", since="8.18")]
Lemma Zodd_bit_value :
forall z:Z, Zeven.Zodd z -> bit_value (Z.odd z) = 1%Z.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Zge_minus_two_power_nat_S :
forall (n:nat) (z:Z),
(z >= - two_power_nat (S n))%Z -> (Zmod2 z >= - two_power_nat n)%Z.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma Zlt_two_power_nat_S :
forall (n:nat) (z:Z),
(z < two_power_nat (S n))%Z -> (Zmod2 z < two_power_nat n)%Z.
End Z_BRIC_A_BRAC.
Section COHERENT_VALUE.
We check that the functions are reciprocal on the definition interval. This uses earlier library lemmas.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma binary_to_Z_to_binary :
forall (n:nat) (bv:Bvector n), Z_to_binary n (binary_value n bv) = bv.
#[deprecated(note="Consider Z.testbit instead", since="8.18")]
Lemma two_compl_to_Z_to_two_compl :
forall (n:nat) (bv:Bvector n) (b:bool),
Z_to_two_compl n (two_compl_value n (Bcons b n bv)) = Bcons b n bv.
#[deprecated(note="Consider Z.mod_small or Z.bits_inj' instead", since="8.18")]
Lemma Z_to_binary_to_Z :
forall (n:nat) (z:Z),
(z >= 0)%Z ->
(z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z.
#[deprecated(note="Consider Z.mod_small with an input offset or Z.bits_inj' instead", since="8.18")]
Lemma Z_to_two_compl_to_Z :
forall (n:nat) (z:Z),
(z >= - two_power_nat n)%Z ->
(z < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z) = z.
End COHERENT_VALUE.